Working Paper in Economics No. 771
Child Human Capital – The Importance of
Parenting Style
CHILD HUMAN CAPITAL—THE IMPORTANCE OF
PARENTING STYLE
Kristian Bolin1 Michael R. Caputo2
1 Department of Economics, Centre for Health Economics, University of Gothenburg Gothenburg, Sweden. email: kristian.bolin@economics.gu.se
2 Department of Economics, University of Central Florida, 4000 Central Florida Blvd. Orlando, Florida 32816-1400. email: mcaputo@bus.ucf.edu
JEL Codes: C73; D15; J24
Keywords: Differential Game Theory; Intertemporal Choice; Human Capital
Abstract
Introduction
The two most important components of human capital are education and health. Child and adolescent (hereafter child) human capital is important largely because it affects that person’s future prospects for health, education, labour-market, and other essential lifetime outcomes. Ample evidence from both the epidemiological and economic literature supports this statement, see, e.g., Campbell et al. (2014), Currie (2009) and Currie and Almond (2011). Investments in child human capital are made both through public efforts, e.g., vaccination programs and compulsory schooling, and intra-family allocation of time and resources. Public and private investments may be offset or reinforced by activities and consumption choices made by children. For instance, alcohol consumption at an early age, illegal drug abuse, and risky sexual behaviour adversely affect the formation of human capital either directly, through physiological mechanisms, or indirectly, by rendering the production of human capital less effective. Other activities and consumption choices may have the opposite effect, examples of which include following stipulated educational curricula and a wholesome diet, exerting appropriate amounts of physical activities, and taking part in non-trivial cultural activities.
resolution. We adopt a different set of definitions of parenting styles by departing from the typical decision-making mechanism used to allocate resources. Thus, our main objective is to examine the importance of different intra-family decision-making mechanisms for the accumulation of child human capital. To this end, we formulate a differential game that accounts for the parent-child interaction and apply three different solution concepts to it, reflecting three different “parenting styles”.
One of the pioneers in the development of economic theories of family decision-making and human capital was Gary Becker (1981, 1993). The focus of his analyses was on gains to family formation assuming cooperation on the part of the family members. Even though there may be strong incentives for honouring cooperatively made decisions about intra-family conduct, few, if any, domestic actions can be controlled by enforceable agreements. Consequently, incentives for strategic behaviour, generally entailing allocations that differ from cooperatively made Pareto efficient allocations, cannot be perfectly curbed. Thus, several authors argue that strategic behaviour needs to be taken into account when analysing intra-family decision making—see, for instance, King (1982), Konrad and Lommerud (1995), Dufwenberg (2002), and Oosterbeek et al., (2003). The survey of economic theories of the family by Bergström (1997) focused on consensus models of the family, and strategic considerations were only briefly discussed. No doubt, this owes to the fact that, at the time, non-cooperative game theory was largely unexplored as a means of studying the family. More recent contributions to this field include parent-child interaction, as in Chang (2009) and Chang et al. (2005), but from the perspectives of children competing for parental resources.
the other hand, when children reach the age when some autonomy is an essential part of eventually becoming an independent adult, but still are dependent on the family for survival, they will be able to behave strategically in relation to the "freedom" offered by parents.
Given that children can only be governed imperfectly, parents must decide how resources are allocated given strategic, selfish and relatively short-sighted child behaviour. This is reminiscent of Becker’s famous Rotten-kid theorem, which implies that a selfish child will cooperate in maximising family welfare if financial incentives are properly tuned. Thus, a parent who faces a related, but more complex, situation of making optimal investments in child human capital during the child’s upbringing may design financial incentives not only as a means of fulfilling altruistic objectives, but also as a tool for influencing child behaviour.
A differential-game approach to investments in child human capital
The core features of the model involve one parent who has a first-mover advantage in the allocation of wealth between investment in child human capital and their own consumption. Given the parent’s allocation and a lifetime budget constraint, the child chooses how much to consume of each of two goods, of which one has a detrimental effect on human capital while the other has a beneficial effect. It is assumed that the child is old enough to make autonomous decisions regarding how to allocate the available resources between the two types of consumptions.
We derive solutions under three different informational structures governing the parent’s and child’s decision making. The first structure assumes that the parent and child make decisions hierarchically under loop information, referred to as the non-evaluating parenting style, thereby yielding an open-loop Stackelberg equilibrium. This means that the leader, i.e., parent, announces their strategy for the entire planning period at the initial time and thus posses a first-mover advantage in this sense, which is fully committed and conditioned only on the clock and exogenous variables. The second assumes that the parent and child make decisions cooperatively under open-loop information, referred to as the cooperative parenting style, thus resulting in a cooperative equilibrium. The cooperative case reflects the situation in which the parent has implemented joint, as opposed to strategic, decision making. And the third assumes that the parent and child make decisions hierarchically under feedback information, referred to as the evaluating parenting style, hence yielding a feedback Stackelberg equilibrium. This is a situation in which a parent allocates resources contingent on time, the accrued stock of human capital, wealth and exogenous variables, and in which they posses a first-mover advantage at each point in time.
The formal structure of the model is as follows. At each point in time,
0,T , the parent derives utility from investments, ( ) 0I , in child human capital, H( ) 0, and their own consumption,( ) 0
Q , and the child derives utility from two types of consumption, Q1( ) 0 and Q2( ) 0. Consumption of Q1 adds to the change in the child’s stock of human capital over time, while consumption of Q2 subtracts from it. The human-capital state equation is
1 2
( ) ( ( ), ( ), ( )) ( )
H f I Q Q H , H(0)H0, (1)
where f( ) is a human-capital production function such that f( ) C(2), with partial derivatives 0 I f , 1 0 Q f , and 2 0 Q
f , and H0 is the initial stock of child human capital.
The following additional assumptions are then adopted:
(A1) Parental (p) and child (c) preferences are represented by the utility functions
(2) ( ( ), ( )) p U Q I C , (2) (2) 1 2 ( ( ), ( )) c U Q Q C , (3) where Uij 0, i p c j, ; 1, 2.
(A2) Parental preferences for the stocks of human capital and financial assets at T, are
represented by the utility function HH T( )AA T( ), whereH 0 and A0.
(A3) Child preferences for the stocks of human capital and financial assets at T, are
represented by the utility function HH T( )AA T( ), where H 0 and A0.
(A4) There are perfect financial markets, which allow lending and borrowing at the continuously compounded interest rate r0.
Assumption (A1) places prototypical restrictions on the instantaneous utility functions, that is, (i) they are twice continuously differentiable and (ii) Q I, and Q Q1, 2are goods. The formulation of the scrap-value functions in stipulations (A2) and (A3) permits the derivation of explicit solutions to the differential game. Assumption (A4) is a standard assumption. It means that financial resources can be transferred between time periods and thus that a lifetime budget constraint applies, as may be seen in what follows. On the other hand, in the absence of financial markets, borrowing and lending is not possible, in which case a different budget constraint would bind at each point in time.
Budget constraint
Given perfect financial markets, the budget constraint faced by both agents asserts that the present value of expenses must be less than or equal to the present value of available resources. More formally, the budget constraint is
1 2
0 0 0 T T - r r I( ) Q( ) Q ( ) Q ( ) e d A yed
, (4)where A0 0 is the initial stock of assets, and y0 is a fixed flow of income. By defining
def ( ) 0 1 2 0 ( ) r ( ) ( ) ( ) ( ) e r t A A e y I t Q t Q t Q t dt
, (5)and then using Leibniz’s rule and the lifetime budget constraint, it follows that
1 2
( ) ( ) ( ) ( ) ( ) ( )
A rA y I Q Q Q , A(0) A0, A T( )0. (6)
The optimal control problems
def 0 0 ( ), ( ) 0 ( , ) max ( ), ( ) p [ ( ) ( )] p T T p p H A Q I V A H U Q I e d H T A T e
, (7)under different informational structures, and subject to the state equation (1), the budget constraint (6), and the non-negativity constraints on the control variables. Similarly, letting cbe the child rate
of time preferences, the child acts as if solving
1 2 def 0 0 1 2 ( ), ( ) 0 ( , ) max ( ), ( ) c [ ( ) ( )] c T T c c H A Q Q V A H U Q Q e d H T A T e
, (8)subject to the state equation (1), the budget constraint (6), and the said non-negativity constraints.
In the cooperative case, the parent and the child behave as if solving
1 2 1 2 def 0 0 0 ( ), ( ) ( ), ( ) ( ( ), ( )) ( ( ), ( )) ( , ) max , ( ) ( ) ( ) ( ) T p c I Q Q Q T H A H A U Q I U Q Q e d V A H H T A T H T A T e
(9)subject to the aforementioned equations.
In the subsequent sections we, first, demonstrate that (i) the open-loop Stackelberg equilibrium is time consistent, (ii) an open-loop Nash equilibrium is a degenerate feedback Nash equilibrium, (iii) an open-loop Stackelberg equilibrium is identical to an open-loop Nash equilibrium, and (iv) a feedback Stackeberg equilibrium is identical to a feedback Nash equilibrium, and, second, derive an optimal control vector for each parenting style. Function arguments are omitted where appropriate.
General properties of noncooperative equilibria
time paths, Q1( ) and Q2( ) according to (8), and (c) the parent chooses investments in child human capital and consumption according to (7), using the child’s reaction functions.
An open-loop Stackelberg equilibrium is not in general time consistent, as shown by Xie (1997) and Dockner et al. (2000). However, the aforesaid authors demonstrated that if the initial values of the follower’s costate variables are not controllable by the leader, then an open-loop Stackelberg equilibrium is time consistent. We now demonstrate that this is the case in the model outlined above.
To that end, define the child’s, i.e., the follower’s, current-value Hamiltonian function by
1 2 1 2 e 1 2 d f 1 2 ( , , , , , ) ( , ) [ ( , , ) ] [ ] c c c c c c A H H A rA y A H Q Q U Q Q f I Q Q H I Q Q Q . (10) where c A and c H
are the child’s costate variables corresponding to the state equations for assets and human capital, respectively. Given assumption (A5), the maximum principle gives the following necessary and sufficient conditions for optimality:
1( 1, 2) 1( , 1, 2) 0 c c c Q H Q A U Q Q f I Q Q , 2( 1, 2) 2( , 1, 2) 0 c c c Q H Q A U Q Q f I Q Q , cH [ c ] Hc ,
Hc( )T
H, [ ] c c A c r A ,
Ac( )T
A. (11)By the implicit function theorem the follower’s reaction functions, say 1
1 ( , , ) c c A H Q R I and 2 2 ( , , ) c c A H
Q R I , exist and are locally C(1).
Observe that the two costate equations are wholly independent of the parent’s control variables, and thus so are the initial values of said costate variables, thereby establishing the ensuing proposition.
Proposition 1: The open-loop Stackelberg equilibrium in the parenting game described by Eqs. (7)
The preceding means that the announced consumption and investment strategies are indeed credible in the eyes of the child. Next, observe that (i) the instantaneous utility functions of the parent and the child are independent of the state variables ( , )A H , (ii) the state equations (1) and (6) are linear in
( ,A H), and (iii) the salvage value functions are linear in ( , )A H . Consequently, the differential game
under consideration meets the definition of a linear state game by Dockner et al. (2000, p. 188). As discussed by Dockner et al. (2000, p. 189), the following proposition is an immediate consequence of this fact.
Proposition 2: If an open-loop Nash equilibrium exists in the differential game defined by Eqs. (1),
(6), (7) and (8), then it is a degenerate feedback Nash equilibrium and, hence, subgame perfect.
Define the parent’s, i.e., the leader’s, current value Hamiltonian function by
def 1 2 1 2 ( , , , , , , , , ( , ( , ( , ( , [ ] , ) ( , ) , ), , ) , ) , ) + c c [ ] p p p p p p p c c c c A H A H H A H A H p c c c c A A H A H p p c H H A c A A H Q I U Q f I R I R I H rA y I Q r R I I R I (12) where p A and p H are the parent’s costate variables corresponding to the state equations for assets
and human capital, respectively, and p A
and p H
are the parent’s costate variables corresponding to the costate equations of the child for assets and human capital. Given assumption (A5), the necessary and sufficient conditions are
[ ] p p H p H , Hp( )T H, [ ] p p A p r A ,
Ap( )T
A,
1 2 1 2 1 1 2 2 1 2 [ ] , ( , , ), ( , , ) ( , , ) , ( , , ), ( , , ) ( , , ) ( , , ) ( , , ) , c H c H c c H H p p H p c H c c c c c c Q A H A H A H c c c c c c p Q A H A H A H H c c c c p A H A H A f I R I R I R I f I R I R I R I R I R I
1 2 1 2 1 1 2 2 1 2 [ ] , ( , , ), ( , , ) ( , , ) , ( , , ), ( , , ) ( , , ) ( , , ) ( , , ) . c A c A c c A A p p A p c A c c c c c c Q A H A H A H c c c c c c p Q A H A H A H H c c c c p A H A H A r f I R I R I R I f I R I R I R I R I R I (13)Now assume that f( ) is jointly linear in all its arguments. This assumption implies the follower’s
reaction functions, and that the necessary and sufficient conditions determining the leader’s open-loop Stackelberg equilibrium are identical to the necessary and sufficient conditions that determine an open-loop Nash equilibrium, thereby establishing the next proposition.
Proposition 3: If f( ) is jointly linear in all its arguments, then an open-loop Stackelberg equilibrium
coincides with an open-loop Nash equilibrium.
The linearity of f( ) therefore implies that the leadership by a parent has no effect on the equilibrium
of the game given open-loop information. This section is brought to a close by deducing one more general result. First, observe that UQp( , )Q I and ( , )
p I
U Q I are independent of (Q Q1, 2), and that
1( 1, 2) c Q U Q Q and 2( 1, 2) p Q
U Q Q are independent of ( , )Q I . Second, note that the two state equations
that the information structure is of the feedback variety. The veracity of the ensuing proposition then follows immediately from footnote 3 and Proposition 2.3 of Rubio (2006).
Proposition 4: If f( ) is jointly linear in all its arguments and the information structure of the game
defined by Eqs. (1), (6), (7), and (8) is of the feedback variety, then a feedback Stackelberg equilibrium coincides with a feedback Nash equilibrium.
Taken together, Proposition 3 and 4 show that the first-mover advantage of a parent vanishes when
( )
f is linear, whether the information structure is open-loop or feedback. In what follows, we
maintain the assumption that f( ) is jointly linear in all its arguments.
Next, we derive explicit solutions associated with the three different parenting styles. To that end, we begin by placing additional stipulations on the differential game.
Additional assumptions regarding the model specification
The following additional assumptions are made:
(A1’) Parental and child preferences are represented by quadratic instantaneous utility functions 2 2 1 1 2 2 ( ( ), ( )) ( ) ( ) ( ) ( ) p U Q
I
Q
Q
I
I
, (14) 2 2 1 1 1 2 1 2 1 2 2 2 ( ( ), ( )) ( ) ( ) ( ) ( ) c U Q Q Q Q Q Q , (15)where q³0 is a parameter reflecting the child’s relative preferences for the two consumption goods.
(A6) Consumption of Q1 adds 1Q1 (10) to, while consumption of Q2 deducts 2Q2 2
( 0) from, the stock of human capital.
(A7) The stock of human capital does not depreciate during the part of the child’s lifecycle considered here.
(A8) The rate of time discounting is zero.
Assumptions (A1’) and (A6) are made in order to derive closed-form solutions, whereas assumption (A7) is made for convenience. By assumption (A5’) complications involving boundary solutions are avoided. Due to the linearity of f( ) and assumptions (A1’) and (A6), the Mangasarian sufficiency
theorem can be invoked and, hence, the optimal control vector is unique. A more general specification, allowing for proportional deterioration of the stock of human capital, would not influence the qualitative content of our results. Likewise, assumption (A8) does not qualitatively influence the equilibria either, as it merely introduces a scaling of the equilibrium amount of child human capital at compared to the case with a strictly positive rate of time discounting.
Given the linearity of ( )f and assumption (A6), the human-capital state equation becomes
H( ) I( ) 1Q1( ) 2Q2( ) . (16)
We start by considering the cooperative family, in which case the allocation derived is by construction Pareto efficient. This solution serves as a benchmark for the two non-cooperative solutions considered.
Cooperative parenting
The parent and the child act as if solving optimization problem (9), subject to Eqs. (6) and (16). The Hamiltonian function is:
T
def 1 2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 2 2 1 1 2 2 1 2 ( , , , , , , , ) , A H H A A H Q I Q Q Q Q I I Q Q Q Q I Q Q rA y Q I Q Q (17)where A and H are the costate variables corresponding to the asset and human capital state
equations, respectively. The necessary and sufficient optimality conditions for a strictly positive control vector are:
1 Q A0, 1 I H A0, 1 1 (1 Q) H A 0 , 2 2 1Q H A0, 0 H , H( )T H H, A r A ,A( )T AA. (18)
Seeing as H( )H H 0 and A( )(AA)er T( ) 0, it follows that the cooperative equilibrium is given by * ( ) ( ) 1 ( H H) ( A A) r T I e , * ( ) ( ) 1 ( A A) r T Q e , * 1 1 ( ) 1( ) 1 1( ) ( ) r T H H A A Q e , * ( ) 2( ) 1 2( ) ( ) r T H H A A Q e . (19)
Child human-capital investment, and the consumption of all three goods, are increasing over time at a rate proportional to the joint preference for the terminal stock of financial assets, AA. This is so
for * 1( )
Q . In contrast, parental consumption and child consumption of the good that is harmful to
human capital are always less than the satiation rate.
The terminal value of child human capital is endogenous, and so must be found by solving the state equation for human capital using the cooperative equilibrium. Its derivation is given in the appendix and yields the proof of the following proposition.
Proposition 5: Given Eq. (19) and the human-capital state equation, the optimal amount of child human capital at T
in the cooperative family is given by:
1 1 1 2 2
0 1 2 1 2 1 2
( ) ( A A) 1 rT 1 1 ( 1)( H H)
H T H r e T . (20)
A few observations about the optimal amount of child human capital for the cooperative equilibrium at the end of the planning horizon are in order. First, note that the strength of the parent’s and the child’s preferences for the child’s human capital at T is positively related to the size of the human
capital stock at that time. This is so because the stronger those preferences are, more is invested and spent on the good with a beneficial effect on human capital, and less is spent on the good with a detrimental effect. Second, the parent’s and the child’s preferences for the stock of financial wealth at
T
might be positively or negatively related to the stock of human capital at the end of the planning horizon. The reason is that the stronger the parent’s or child’s preferences for terminal wealth the less is spent on all goods that influence the stock, and that the beneficial effect of reducing
2
Q may or may not outweigh the negative effect of reducing I and Q1. Third, the extent to which
the detrimental effect that good two has on the child’s stock of human capital (2)at the end of the planning horizon is given by
if 2 2( H H)1. The intuition behind this somewhat counterintuitive result is that the effect on
the stock of an increase in 2works through Q2 only, and that the positive indirect effect, *
2 Q2 2
, outweighs the negative direct effect, * 2
Q
.
The non-evaluating parenting style
By Proposition 3, the equilibrium resulting from the non-evaluating parenting style is derived by computing the open-loop Nash equilibrium of the game. Thus, the Hamiltonian functions associated with the child’s and the parent’s optimization problems are
def 1 2 1 2 1 2 1 2 2 2 1 2 1 2 1 1 2 2 ( , , , , , ) c c c c A H H c A Q Q Q Q I rA Q A H Q Q Q Q y I Q Q (21)
def 2 2 2 1 2 1 1 1 2 2 1 1 2 2 ( , , , , , , , , , ) . p p p p p p A H A H H p A Q Q I I A H Q I Q Q I Q Q Q rA y I Q Q (22)The necessary and sufficient conditions for optimality for a strictly positive child control vector are
1 1
1 0 c c H A Q
, 2 2 1Q Hc Ac 0, 0 c H ,
Hc( )T
H, ,
Ac( )T
A, (23)And those for the parent are
1 1 ( ) 1( ) 1 1 S r T H A Q e , ( ) 2( ) 1 2 S r T H A Q e . (25)
As was the case in the cooperative family, investment and all consumption increase over time in the non-evaluating family. In the present case, however, the rate of increase is proportional to each agent’s marginal valuation of the terminal stock of financial assets, rather than to the family’s marginal valuation of the terminal stock, implying that the rate of increase is smaller in the non-evaluating family than in the cooperative family. Deductions concerning the position of each control trajectory in relation to unity are analogous to those made in the cooperative case, with the difference that in this case the relative strength of each agent’s preferences for the terminal stocks determines the position of each trajectory. So, for instance, if AH, then
* ( ) 1
I for all
0,T .Again, the stock of child human capital at Tin the non-evaluating family is obtained by solving
the human-capital state equation (16) using the above open-loop Stackelberg equilibrium. The result is given in proposition 6 and proven in the appendix.
Proposition 6: Given Eq. (25) and the human-capital state equation, the equilibrium amount of child human
capital at Tin the non-evaluating family is given by:
1 1 1 0 2 1 1 1 2 2 ( ) ( ) ( 1) 1 (1 ) (1 ) S rT A A H H H H T H r e T. (26) The following observations are noteworthy. First, the more the parent prefers child human capital at time T , the larger is the stock of it at that time. This is so, since as H increases, more is invested
in child human capital during the family’s lifetime. The opposite is true regarding the strength of the parent’s preferences for terminal wealth A T( ). Second, as was the case in the cooperative family, the
child’s preferences for A T( ) might be positively or negatively related to H TS( ), as they are negatively linked to *
1
of the child’s preferences for H T( ) is positively related to S( )
H T , since His positively related to
1 ( )
S
Q and negatively related to Q2S( ) . And fourth, the effect of an increase in j2 on the child’s terminal human capital is given by 1
2 2 ( ) ( 1) (2 1) S rT A H H T r e T , which may be
positive or negative, depending on the magnitude of A and H. But if 2 2 H 1, then an increase
in the marginal damage to human capital as a result of the child’s consumption of good two leads to a larger terminal stock of human capital.
The accumulation of child human capital differs between the two preceding parenting styles according to the following proposition, as may be seen by comparing Eqs. (20) and (26).
Proposition 7: The difference *
( ) S( ) H T H T is given by 1 1 1 2 2 2 1 1 2 ( ) S( ) A( ) A rT 1 H( ) H H T H T r e T . (27)
The amount of child human capital accumulated at the end of the family’s lifetime may be larger or smaller under cooperative parenting compared to non-evaluating parenting, depending on the preferences of the parent and the child, and the magnitude of the effects of the child’s consumption on the evolution of their capital stock. For example, the larger is the magnitude of the detrimental human-capital effect of Q2, the larger is the difference
*
( ) S( )
H T H T . So, cooperative parenting
will lead to more child human capital than non-evaluating parenting when 2 is (sufficiently) large. The one situation in which the non-evaluating parenting style leads to more child human capital than the cooperative parenting style involves sufficiently strong child preferences for the stock of financial assets at T, which, in the cooperative case, will be incorporated directly in the family’s decision
The evaluating parenting style
In this section, we assume that the parent posses a first-mover advantage and has adopted a feedback strategy. This means that the parent’s decisions about investment and consumption at each point in time are made based upon the accrued values of the stocks of financial assets and human capital. As demonstrated by Dockner et al. (2000), the feedback Stackelberg equilibrium can be derived using dynamic programming. More specifically, in our case we derive an explicit solution by conjecturing specific functional forms of the value functions of the parent and the child and, then, by employing the method of undetermined coefficients to the two Hamilton-Jacobi-Bellman (HJB) equations. Recall that the feedback Stackelberg and the Nash equilibria coincide, by Proposition 4. We thus derive the feedback Nash equilibrium.
By Theorem 3.1 of Dockner et al. (2000) the HJB equation corresponding to the parent’s optimization problem (7) is given by
1 2 1 2
1 1 2 2 1 2 2 2 , ( , , ) max p p p H A I Q V A H I I Q Q V I Q Q V rA y Q I Q Q , (28)with terminal condition Vp( , ( ),T A T H T( ))AA T( )a H TH ( ). Similarly, the HJB equation corresponding to the child’s problem (8) is given by
1 2 2 2 1 1 1 2 1 2 2 2 1 1 2 2 , 1 2 ( , , ) max c H c c Q Q A Q Q Q Q V I Q Q V A H V rA y Q I Q Q , (29)with terminal condition V T A T H Tc( , ( ), ( ))AA T( )HH T( ).
Given the functional forms in the two optimization problems, reasonable conjectures about the parent’s and child’s value function are
In order for the conjectured value functions to satisfy the corresponding HJB equations, the time dependent coefficients must be determined so that the value functions satisfy the corresponding HJB equations (28) and (29), for all (feasible) values of ( , , A H). This amounts to (i) maximizing the
right-hand sides of the HJB equations with respect to the control variables and solving for the control variables, (ii) substituting the solutions from (i) back into to each HJB equation, (iii) substituting the partial derivatives of the conjectured value functions in Eqs. (30) and (31) in the HJB equations, and (iv) equating the coefficients on ( ,A H,1) on the right- and left-hand sides of the HJB equations. This
procedure leads to a six-equation system of ordinary differential equations, the solution to which is given in the following proposition—the proof is outlined in the appendix.
Proposition 8: Let * denote a solution to a differential equation for a coefficient in Eqs. (30) and (31). Then, given
the functional forms in Eqs. (14)–(16), Eqs. (30) and (31) are value functions for the corresponding optimal control problems (7) and (8), if * * ( ) [ ( ),a b ( )] [ H, Aer T ], (32) * * ( ) [d ( ), ( )] e [ H, Aer T ], (33) where * ( )
c and f*( ) satisfy the differential equations c h t( , )x and f g t( , )x , where xis the vector
of parameters (see the appendix).
The evaluating family equilibrium amount of child human-capital investment and consumption may then be obtained by using the conjectured value functions and (32) and (33). These calculations are outlined in the appendix, and the result is summarized in the following corollary.
Corollary 1: The feedback Stackelberg equilibrium resulting from the value functions derived in Proposition 8, is
Corollary 1 shows that the evaluating parenting and non-evaluating parenting styles result in identical decisions. What is more, because the open-loop Stackelberg equilibrium, open-loop Nash equilibrium, feedback Nash Equilibrium, and feedback Stackelberg equilibrium all coincide with each other, there is no first mover advantage for the parent whether the parent commits to a plan of action at the outset of the game or does not commit at all.
Summary and conclusion
References
Becker, Gary S. A treatise on the family. Cambridge, Massachusetts: Harvard University Press, 1981. Becker, Gary S. Human Capital: A Theoretical and Empirical Analysis, with Special Reference to Education. Columbia University Press, 1993.
Bergström, T. A survey of theories of the family, in (M. Rosenzweig and O. Stark, eds.) Handbook of Population Economics, pp. 21-74, New York: North Holland, 1997.
Campbell, F., Conti, G., Heckman, J.J., Moon, S.H., Pinto, R., Pungello, E. and Pan, Y. Early childhood investments substantially boost adult health, Science, 2014;343:1478-1485.
Chang, YM. Strategic altruistic transfers and rent seeking within the family. Journal of Population
Economics, 2009;22:1081-1098.
Chang, Y.M., Weisman, D.L. Sibling rivalry and strategic parental transfers. Southern Economic Journal, 2005; 71(4):821-836.
Currie, J. Healthy, wealthy, and wise: socioeconomic status, poor health in childhood, and human capital development, Journal of Economic Literature, 2009;47(1):87-122.
Currie, J., Almond, D. Human capital development before age five, in (O. Ashenfelter and D. Card) Handbook of Labor Economics, pp. 1315-1486, North Holland, 2011.
Dockner, E., Jorgensen, S., Van Long, N., Sorger, G. Differential Games in Economics and Management
Science, Cambridge: Cambridge University Press, 2000.
Doepke M, Zilibotti F. Parenting with Style: Altruism and Paternalism in Intergenerational Preference Transmission. Econometrica, 2017;85(5):1331–1371.
Dufwenberg, M. Marital investments, time consistency and emotions. Journal of Economic Behavior and
Organization, 2002;48:57–69.
Ewald C.O., McNamara J., Houston A. Parental care as a differential game: A dynamic extension of the Houston–Davies game. Applied Mathematics and Computation, 2007;190:1450–1465.
Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.A. Optimal control of nonlinear processes – with applications in drugs, corruption and terror. Spinger, 2008.
Hori, K., Shibata, A. Dynamic Game Model of Endogenous Growth with Consumption Externalities.
Journal of Optimization Theory and Applications, 2010;145: 93–107
King, A. Human capital and the risk of divorce: an asset in search of a property right. Southern
Economic Journal, 1982;49(2):536–541.
Konrad, K., Lommerud, K. Non-Cooperative families. Scandinavian Journal of Economics, 1995;97:581– 601.
Long, N.V. Dynamic Games in the Economics of Natural Resources: A Survey. Dynamic Games and
Applications, 2011;1:115–148
Novak, A.J., Feichtinger, G., Leitmann, G. A Differential Game Related to Terrorism: Nash and Stackelberg Strategies. Journal of Optimization Theory and Applications, 2010; 144:533–555.
Oosterbeek, H., Sonnemans, J., van Velzen, S. The need for marriage contracts: an experimental study. Journal of Population Economics, 2003;16:431–453.
Rubio, S.J. On coincidence of Feedback Nash Equilibria and Stackelberg equilibria in Economic Applications of Differential Games. Journal of Optimization Theory and Applications, 2006;128:203-221. Xie, D. On Time Inconsistency: A Technical Issue in Stackelberg Differential Games. Journal of
Appendix
Proof of Proposition 5
Substituting the variables in the human-capital state equation (16) for the optimal control vector given by (19) yields ( ) 1 1 ( ) 1 1 ( ) 2 2 ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) . r T r T H H A A H H A A r T H H A A H e e e
This differential equation is of the form ( )
( ) r T H a be , where
def 1 1 1 2 2 1 ( H H) 1 ( H H) 1 ( H H) , a def 1 1 2 ( A A) ( A A) ( A A). b The general solution is 1 ( )
( ) r T
H a r be c, where c is a constant of integration. The initial
condition H(0)H0 gives 1 0 rT cH r be . Hence H( ) H0r b e1 ( rT er T( ))a and * 1 0 ( ) ( rT 1)
H T H r b e aT . Inserting the expressions for aand b in H T*( ) and rearranging terms gives Eq. (20).
Proof of Proposition 6
Substituting the open-loop Stackelberg equilibrium values given by (25) in the human-capital state equation (16) yields
( ) 1 1 ( ) ( )
1 1 2 2
( ) 1 H A r T 1 H A r T 1 H A r T .
H e e e
Proceeding analogously to the proof of Proposition 2 gives Eq. (26).
Proof of Proposition 8
1 I VHp VAp 0, 1 Q VAp 0,
1 1
1 0 c c H A Q V V , 2 2 1Q VHcVAc 0.Next, solve for the four control variables from the preceding optimality conditions, substitute them in each HJB equation, and then substitute the partial derivatives of the conjectured value functions in Eqs. (30) and (31) in the HJB equations to arrive at the form of the HJB equations used to find the six coefficients in Eqs. (30) and (31). For the parent, the HJB equation becomes
1 2
2 1 2 1 1 2 1 2 1 1 2 2 2 1 1 2 ( ) ( ) ( ) 4 ( ) 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) a H b A c y b a a b rb A b a e b b d a e a d where
( )
( ) 1 2 2 2 ( ) 1 2 1 ( ) ( ) 1 ( ) 2 1 2 1 ( ) 1 2 2 2 1 1 2 4 1 1 1 ( ) r T r T A H H A r T r T r T r T A H A A H A r T H A H H y e e e e e e c e and for the child the HJB equation is
1 1 2 2 1 1 1 2 2 1 2 ( ) ( ) ( ) 1 4 ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) 2 ( ) ( ) , ( ) ( ) ( ) ( ) ( ) ( ) d H e A f y e d re A d e e e b a e d b d e where
( )
1 1 2 2 1 ( ) 1 1 2 ( ) ( ) ( ) 2 1 2 ( ) ( ) 1 4 1 ( ) 1 2 r T A H r T r T r T r T H A A A A r T r T H A H A H H y e f e e e e e e In order for the conjectured value functions to satisfy the corresponding HJB equations for all ( , , A H), the coefficients on ( , ,1)A H must be identical on each side of both HJB equations.
